Bonchi et al. [1]} showed that the final \(\mathcal {L}\) -coalgebra
is defined on the set of all weighted languages \(K^{A^\ast }\)  as follows.
Consider the structure \((K^{A^\ast },\epsilon ,d)\)  with the output map \(\epsilon \) 
and the transition map \(d\)  where \(\epsilon :K^{A^\ast } \rightarrow K\) , called the
empty map, is defined by \(\epsilon (\sigma )=\sigma (\epsilon )\)  and
\(d:K^{A^\ast } \rightarrow (K^{A^\ast })^A\)  is defined by \(d(\sigma )(a)= \sigma _a\)  where
\(\sigma _a: A^\ast \rightarrow K\)  is the \(a\) -derivative of \(\sigma \) :
\( \sigma _a(w)=\sigma (aw).\) 
